Introduction

This tutorial provides a comprehensive introduction to a modeling and simulation workflow in Pumas. This tutorial will not get into the details of Pumas specifics, but instead provide a narrative on the lines of a regular workflow in our day to day work, with brevity where required to allow a broad overview. Wherever possible, cross-references will be provided to documentation and detailed examples that provide deeper insight into a particular topic.

As part of this workflow you will be introduced to various aspects such as

1. Data wrangling in Julia

2. Exploratory analysis in Julia

3. Continuos and discrete data non-linear mixed effects modeling in Pumas

4. Model comparison routines, post-processing, validation etc.

The Study and Design

CTMNopain is a novel anti-inflammatory agent under preliminary investigation. A dose-ranging trial was conducted comparing placebo with 3 doses of CTMNopain (5mg, 20mg and 80 mg QD). The maximum tolerated dose is 160 mg per day. Plasma concentrations (mg/L) of the drug were measured at 0, 0.5, 1, 1.5, 2, 2.5, 3-8 hours.

Pain score (0=no pain, 1=mild, 2=moderate, 3=severe) were obtained at time points when plasma concentration was collected. A pain score of 2 or more is considered as no pain relief.

The subjects can request for remedication if pain relief is not achieved after 2 hours post dose. Some subjects had remedication before 2 hours if they were not able to bear the pain. The time to remedication and the remedication status is available for subjects.

The pharmacokinetic dataset can be viewed here, and later in the tutorial, we show you how to download for use. The remedication dataset will be used in part-2 of the tutorial.

Setup

Load libraries

Two libraries provide the workhorse functionality in the Pumas ecosystem that we will load:

using Pumas
using PumasUtilities

The libraries below are good add-on's that provide additional functionality.

using Weave
using Random
using CSV
using HTTP
using Chain
using Latexify
using CairoMakie

If you get a message that these packages are not available in your system, please add them as follows e.g.

using Pkg
Pkg.add("Weave")

For the purpose of this tutorial, we set all plots to be non-interactive. While working in a interactive environment using .jl files, it is perhaps efficient to use leave the option to the default value of true. More about these options can be found about in our documentation.

interactive!(false)

false

Data Wrangling

We start by reading in the dataset and making some quick summaries.

Note: As a general convention during this example, dataframes will be referred by ending with the name of the variable with df and the Population version of that dataframe will be without the df to avoid confusion.

f = CSV.File(HTTP.get("https://raw.githubusercontent.com/PumasAI/PumasTutorials.jl/master/data/intro/pk_painscore.csv").body, missingstrings=["", "NA", "."])
pkpain_df = DataFrame(f)

Let's filter out the placebo data as we don't need that for the PK analysis.

pkpain_noplb_df = filter(x -> !(occursin.("Placebo", x.ARM)), pkpain_df)

1,440 rows × 6 columns

	ARM	ID	TIME	CONC	PAINRELIEF	DOSE
	String	Int64	Float64	Float64	Int64	Int64
1	A20_0_at2h	1	0.0	0.0	0	20
2	A20_0_at2h	1	0.5	1.15578	1	20
3	A20_0_at2h	1	1.0	1.37211	1	20
4	A20_0_at2h	1	1.5	1.30058	1	20
5	A20_0_at2h	1	2.0	1.19195	1	20
6	A20_0_at2h	1	2.5	1.13602	1	20
7	A20_0_at2h	1	3.0	0.873224	1	20
8	A20_0_at2h	1	4.0	0.739963	1	20
9	A20_0_at2h	1	5.0	0.600143	0	20
10	A20_0_at2h	1	6.0	0.425624	1	20
11	A20_0_at2h	1	7.0	0.363418	1	20
12	A20_0_at2h	1	8.0	0.304177	1	20
13	A80_0_at2h	2	0.0	0.0	0	80
14	A80_0_at2h	2	0.5	4.93492	1	80
15	A80_0_at2h	2	1.0	4.56849	1	80
16	A80_0_at2h	2	1.5	4.23436	1	80
17	A80_0_at2h	2	2.0	3.35444	1	80
18	A80_0_at2h	2	2.5	2.70046	1	80
19	A80_0_at2h	2	3.0	2.38586	1	80
20	A80_0_at2h	2	4.0	1.84095	1	80
21	A80_0_at2h	2	5.0	1.64535	1	80
22	A80_0_at2h	2	6.0	1.36486	1	80
23	A80_0_at2h	2	7.0	1.20802	1	80
24	A80_0_at2h	2	8.0	0.983679	1	80
⋮	⋮	⋮	⋮	⋮	⋮	⋮

Analysis

Non-compartmental analysis

Let's begin by performing a quick NCA of the concentration time profiles and view the exposure changes across doses. The input data specification for NCA analysis requires the presence of a route column and an amount column that specifies the dose. So, let's add that in.

#adding route variable
pkpain_noplb_df[!,:route] .= "ev"
# creating an `amt` column
pkpain_noplb_df[!,:amt] .= ifelse.(pkpain_noplb_df.TIME .== 0, pkpain_noplb_df.DOSE, missing)

Now, we map the data variables to the read_nca function that prepares the data for NCA analysis.

pkpain_nca = read_nca(pkpain_noplb_df,
                      id = :ID,
                      time = :TIME,
                      amt = :amt,
                      observations = :CONC,
                      group = [:DOSE],
                      route = :route)

Now that we mapped the data in, let's visualize the concentration vs time plots for a few individuals

observations_vs_time(pkpain_nca[1:4], separate=true, 
                    columns = 4, rows = 4)[1]

or you can view the summary curves by dose group as passed in to the group argument in read_nca

f = summary_observations_vs_time(pkpain_nca, 
                                  columns=2, rows=3,
                                  fontsize = 10)
#how to convert to log scale - cc Michael
f[1]

A full NCA Report is now obtained for completeness purposes using the run_nca function, but later we will only extract a couple of key metrics of interest.

pk_nca = run_nca(pkpain_nca, sigdig=3)
first(pk_nca.reportdf, 10)

10 rows × 40 columns (omitted printing of 31 columns)

	id	DOSE	doseamt	tlag	tmax	cmax	tlast	clast	clast_pred
	Int64	Int64	Int64	Float64	Float64	Float64	Float64	Float64	Float64
1	6	5	5	0.0	1.0	0.316115	8.0	0.110616	0.107079
2	8	5	5	0.0	0.5	0.335649	8.0	0.0962241	0.0960789
3	9	5	5	0.0	0.5	0.538566	8.0	0.0430812	0.0387015
4	12	5	5	0.0	1.0	0.215302	8.0	0.0664953	0.0653076
5	17	5	5	0.0	1.0	0.190482	8.0	0.0558857	0.0553108
6	24	5	5	0.0	1.0	0.264185	8.0	0.0246395	0.0236468
7	29	5	5	0.0	0.5	0.353126	8.0	0.0376847	0.0366889
8	32	5	5	0.0	0.5	0.311369	8.0	0.151186	0.143299
9	38	5	5	0.0	0.5	0.295493	8.0	0.0771599	0.0758813
10	40	5	5	0.0	0.5	0.44879	8.0	0.0650122	0.0645857

We can look at the NCA fits for some subjects.

f = subject_fits(pk_nca,
                 separate = true, 
                 columns = 4, rows = 3, fontsize = 12)

f[12]

As CTMNopain's effect maybe mainly related to maximum concentration (cmax) or area under the curve (auc), we present some summary statistics using the summarize function from NCA.

strata = [:DOSE]
parms = [:cmax, :aucinf_obs]
output = summarize(pk_nca.reportdf; stratify_by = strata, parameters = parms)

6 rows × 9 columns (omitted printing of 2 columns)

	DOSE	parameters	extrema	geomean	geomeanCV	geostd	mean
	Int64	String	Tuple…	Float64	Float64	Float64	Float64
1	5	aucinf_obs	(0.9137, 3.39947)	1.53397	86.6964	1.32989	1.59819
2	5	cmax	(0.190482, 0.538566)	0.345132	374.621	1.29294	0.356087
3	20	aucinf_obs	(2.77495, 14.1139)	6.01975	23.4821	1.41356	6.3764
4	20	cmax	(0.933113, 2.70061)	1.43399	88.0787	1.26304	1.47354
5	80	aucinf_obs	(13.7102, 49.122)	28.2973	4.74071	1.34149	29.5023
6	80	cmax	(3.29685, 8.47195)	5.64125	22.2948	1.25771	5.78671

The statistics printed above are the default, but you can pass in your own statistics using the stats = [] argument to the summarize function.

We can look at a few parameter distribution plots.

f = parameters_vs_group(pk_nca, :cmax)[1]

Dose normalized PK parameters, cmax and aucinf were essentially dose proportional between for 5 mg, 20 mg and 80 mg doses. Based on visual inspection of the concentration time profiles as seen earlier, CTMNopain exhibited monophasic decline, and perhaps a one compartment model best fits the PK data.

Pharmacokinetic modeling

As seen from the plots above, the concentrations decline monoexponentially. We will evaluate both one and two compartment structural models to assess best fit. Further, different residual error models will also be tested.

We will use the results from NCA to provide us good initial estimates. The mean clearance is 3.29, the mean volume is 16.45 and a good initial estimate for absorption rate as obtained by $0.693/(tmax/4)$ is 3.85

Data preparation for modeling

PumasNDF requires the presence of evid and cmt columns in the dataset.

pkpain_noplb_df[!, :evid] .= ifelse.(pkpain_noplb_df.TIME .== 0, 1, 0)
pkpain_noplb_df[!, :cmt] .= ifelse.(pkpain_noplb_df.TIME .== 0, 1, 2)
pkpain_noplb_df[!, :cmt2] .= 1 # for zero order absorption

Further, observations at time of dosing, i.e., when evid = 1 have to be missing

pkpain_noplb_df[!, :CONC] .= ifelse.(pkpain_noplb_df.evid .== 1, missing, pkpain_noplb_df.CONC)

The dataframe is now converted to a Population using read_pumas. Note that both observations and covariates are required to be an array even if it is one element.

pkpain_noplb = read_pumas(pkpain_noplb_df,
                          id           = :ID,
                          time         = :TIME,
                          amt          = :amt,
                          observations = [:CONC],
                          covariates   = [:DOSE],
                          evid         = :evid,
                          cmt          = :cmt)

Now that the data is transformed to a Population of subjects, we can explore different models.

One-compartment model

pk_1cmp = @model begin
  @metadata begin
    desc = "One Compartment Model"
    timeu = u"hr"
  end
  @param begin
    "Clearance (L/hr)"
    tvcl ∈ RealDomain(lower = 0, init = 3.2)
    "Volume (L)"
    tvv  ∈ RealDomain(lower = 0, init = 16.4)
    "Absorption rate constant (h-1)"
    tvka ∈ RealDomain(lower = 0, init = 3.8)
    """
    - ΩCL
    - ΩVc
    - ΩKa
    """
    Ω    ∈ PDiagDomain(init = [0.04,0.04,0.04])
    "Proportional RUV"
    σ_p  ∈ RealDomain(lower = 0.0001, init = 0.2)
  end
  @random begin
    η ~ MvNormal(Ω)
  end
  @covariates begin
    "Dose (mg)" 
    DOSE
  end
  @pre begin
    CL = tvcl * exp(η[1])
    Vc = tvv * exp(η[2])
    Ka = tvka * exp(η[3])
  end
  @dynamics Depots1Central1
  @derived begin
    cp := @. Central/Vc
    """
    CTMx Concentration (ng/mL)
    """
    CONC ~ @. Normal(cp, abs(cp)*σ_p)
  end
end

PumasModel
  Parameters: tvcl, tvv, tvka, Ω, σ_p
  Random effects: η
  Covariates: DOSE
  Dynamical variables: Depot, Central
  Derived: CONC
  Observed: CONC

Before going to fit the model, let's evaluate some helpful steps.

1. Simulation to check appropriateness of data and model

#zero out the random effects
etas = zero_randeffs(pk_1cmp, 
              init_params(pk_1cmp),
              pkpain_noplb)

120-element Vector{NamedTuple{(:η,), Tuple{FillArrays.Zeros{Float64, 1, Tup
le{Base.OneTo{Int64}}}}}}:
 (η = 3-element Zeros{Float64},)
 (η = 3-element Zeros{Float64},)
 (η = 3-element Zeros{Float64},)
 (η = 3-element Zeros{Float64},)
 (η = 3-element Zeros{Float64},)
 (η = 3-element Zeros{Float64},)
 (η = 3-element Zeros{Float64},)
 (η = 3-element Zeros{Float64},)
 (η = 3-element Zeros{Float64},)
 (η = 3-element Zeros{Float64},)
 ⋮
 (η = 3-element Zeros{Float64},)
 (η = 3-element Zeros{Float64},)
 (η = 3-element Zeros{Float64},)
 (η = 3-element Zeros{Float64},)
 (η = 3-element Zeros{Float64},)
 (η = 3-element Zeros{Float64},)
 (η = 3-element Zeros{Float64},)
 (η = 3-element Zeros{Float64},)
 (η = 3-element Zeros{Float64},)

simpk = simobs(pk_1cmp, 
              pkpain_noplb, 
              init_params(pk_1cmp), etas)
f = sim_plot(pk_1cmp,
            simpk)[1]
f

Our NCA based initial guess on the parameters seem to work well.

Lets change the initial estimate of a couple of the parameters to evaluate the sensitivity.

pkparam = (init_params(pk_1cmp)..., tvka=2, tvv = 10)

(tvcl = 3.2, tvv = 10, tvka = 2, Ω = [0.04 0.0 0.0; 0.0 0.04 0.0; 0.0 0.0 0
.04], σ_p = 0.2)

simpk = simobs(pk_1cmp,
               pkpain_noplb,
               pkparam, etas)
f = sim_plot(pk_1cmp,
            simpk)[1]
f                  
#add observations cc Michael

Changing the tvka and decreasing the tvv seemed to make an impact and observations go through the simulated lines.

To get a quick ballpark estimate of your PK parameters, we can do a NaivePooled analysis. Below we test the NaivePooled approach

NaivePooled

pkfit_np = fit(pk_1cmp,
               pkpain_noplb,
               init_params(pk_1cmp),
               Pumas.NaivePooled(),
               omegas = (:Ω,))

coefficients_table(pkfit_np)

7 rows × 3 columns

	Parameter	Description	Estimate
	String	Abstrac…	Float64
1	tvcl	Clearance (L/hr)	3.01
2	tvv	Volume (L)	14.1
3	tvka	Absorption rate constant (h-1)	44.2
4	Ω₁,₁	ΩCL	NaN
5	Ω₂,₂	ΩVc	NaN
6	Ω₃,₃	ΩKa	NaN
7	σ_p	Proportional RUV	0.33

The final estimates from the NaivePooled approach seem reasonably close to our initial guess from NCA, except for the tvka parameter. We will stick with our initial guess

One way to be cautious before going into a complete fiting routine is to evaluate the likelihood of the individual subjects given the initial parameter values and see if anyone pops out as unreasonable. There are a few ways of doing this:

• check the loglikelihood subject wise

• check if there any influential subjects

Below, we are basically checking if the initial estimates for any subject are way off that we are unable to compute the initial loglikelihood.

lls = []
for subj in pkpain_noplb
  push!(lls,loglikelihood(pk_1cmp,
                   subj,
                   pkparam,
                   Pumas.FOCE()))
end
hist(lls; bins = 10, normalization = :none, color = (:black, 0.5))

The distribution of the loglikelihood's suggest no extreme outliers.

A more convenient way is to use the findinfluential function that provides a list of k top influential subjects

influential_subjects = findinfluential(pk_1cmp,
                   pkpain_noplb,
                   pkparam,
                   Pumas.FOCE())

5-element Vector{Tuple{String, Float64}}:
 ("148", 16.659658856844857)
 ("135", 16.648985190076342)
 ("156", 15.959069556607492)
 ("159", 15.441218240496486)
 ("149", 14.715134644119516)

Now that we have a good handle on our data, lets go ahead and fit a population model

pkfit_1cmp = fit(pk_1cmp,
                 pkpain_noplb,
                 pkparam,
                 Pumas.FOCE(),
                 constantcoef = (tvka = 2,))

infer(pkfit_1cmp)

Asymptotic inference results using sandwich estimator

Successful minimization:                      true

Likelihood approximation:               Pumas.FOCE
Log-likelihood value:                    1231.1881
Number of subjects:                            120
Number of parameters:         Fixed      Optimized
                                  1              6
Observation records:         Active        Missing
    CONC:                      1320              0
    Total:                     1320              0

-----------------------------------------------------------------
        Estimate           SE                    95.0% C.I.
-----------------------------------------------------------------
tvcl     3.1642          0.08662         [ 2.9944  ;  3.334  ]
tvv     13.288           0.27482         [12.749   ; 13.827  ]
tvka     2.0             NaN             [  NaN    ;   NaN     ]
Ω₁,₁     0.084939        0.011021        [ 0.063338;  0.10654]
Ω₂,₂     0.048566        0.0063493       [ 0.036121;  0.06101]
Ω₃,₃     5.5812          1.2198          [ 3.1905  ;  7.9719 ]
σ_p      0.10093         0.0057197       [ 0.089719;  0.11214]
-----------------------------------------------------------------

Notice that tvka is fixed to 2 as we don't have a lot of information before tmax. From the results above, we see that the parameter precision for this model is reasonable.

Just to be sure, let's fit a 2-compartment model and evaluate

pk_2cmp = @model begin
  @param begin
    "Clearance (L/hr)"
    tvcl ∈ RealDomain(lower = 0, init = 3.2)
    "Central Volume (L)"
    tvv  ∈ RealDomain(lower = 0, init = 16.4)
    "Peripheral Volume (L)"
    tvvp ∈ RealDomain(lower = 0, init = 10)
    "Distributional Clearance (L/hr)"
    tvq  ∈ RealDomain(lower = 0, init = 2)
    "Absorption rate constant (h-1)"
    tvka ∈ RealDomain(lower = 0, init = 1.3)
    """
    - ΩCL
    - ΩVc
    - ΩKa
    - ΩVp
    - ΩQ
    """
    Ω    ∈ PDiagDomain(init = [0.04,0.04,0.04, 0.04, 0.04])
    "Proportional RUV"
    σ_p  ∈ RealDomain(lower = 0.0001, init = 0.2)
  end
  @random begin
    η ~ MvNormal(Ω)
  end
  @covariates begin
    "Dose (mg)"
    DOSE
  end
  @pre begin
    CL = tvcl * exp(η[1])
    Vc = tvv *  exp(η[2])
    Ka = tvka * exp(η[3])
    Vp = tvvp * exp(η[4])
    Q = tvq * exp(η[5])
  end
  @dynamics Depots1Central1Periph1
  @derived begin
    cp := @. Central/Vc
    """
    CTMx Concentration (ng/mL)
    """
    CONC ~ @. Normal(cp, cp*σ_p)
  end
end

PumasModel
  Parameters: tvcl, tvv, tvvp, tvq, tvka, Ω, σ_p
  Random effects: η
  Covariates: DOSE
  Dynamical variables: Depot, Central, Peripheral
  Derived: CONC
  Observed: CONC

pkfit_2cmp = fit(pk_2cmp,
                 pkpain_noplb,
                 init_params(pk_2cmp),
                 Pumas.FOCE(),
                 constantcoef = (tvka = 2,))

The 2 compartment model has a much lower objective function compared to the 1 compartment. Lets compare the estimates from the 2 models using the compare_estimate function.

compare_estimates(;pkfit_1cmp, pkfit_2cmp)

11 rows × 3 columns

	parameter	pkfit_1cmp	pkfit_2cmp
	String	Float64?	Float64?
1	tvcl	3.16419	2.81378
2	tvv	13.2881	11.0046
3	tvka	2.0	2.0
4	Ω₁,₁	0.0849389	0.102669
5	Ω₂,₂	0.0485659	0.0607756
6	Ω₃,₃	5.58117	1.20116
7	σ_p	0.100929	0.0484049
8	tvvp	missing	5.53997
9	tvq	missing	1.51591
10	Ω₄,₄	missing	0.423493
11	Ω₅,₅	missing	0.244732

We perform a likelihood ratio test to compare the two nested models. The test statistic and the P-value clearly indicate that a 2 compartment model is better.

lrtest(pkfit_1cmp, pkfit_2cmp)

Statistic:          1200.0
Degrees of freedom:      4
P-value:               0.0

We should also compare the other metrics and statistics, such ηshrinkage, ϵshrinkage, aic, bic using the metrics_table function.

@chain metrics_table(pkfit_2cmp) begin
  leftjoin(metrics_table(pkfit_1cmp), on = :Metric, makeunique = true)
  rename!(:Value => :pk2cmp, :Value_1 => :pk1cmp)
end

11 rows × 3 columns

	Metric	pk2cmp	pk1cmp
	String	Float64	Float64?
1	Estimation Time	4.12	2.84
2	LogLikelihood (LL)	1830.0	1230.0
3	-2LL	-3660.0	-2460.0
4	AIC	-3640.0	-2450.0
5	BIC	-3590.0	-2420.0
6	(η-shrinkage) η₁	0.0367	0.0157
7	(η-shrinkage) η₂	0.0469	0.0402
8	(η-shrinkage) η₃	0.516	0.733
9	(ϵ-shrinkage) CONC	0.185	0.105
10	(η-shrinkage) η₄	0.287	missing
11	(η-shrinkage) η₅	0.154	missing

We next generate some goodness of fit plots to compare which model is performing better. To do this, we first inspect the diagnostics of our model fit.

res_inspect_1cmp = inspect(pkfit_1cmp)
res_inspect_2cmp = inspect(pkfit_2cmp)

gof_1cmp = goodness_of_fit(res_inspect_1cmp, fontsize = 12)[1]
gof_1cmp

gof_2cmp = goodness_of_fit(res_inspect_2cmp, fontsize = 12)[1]
gof_2cmp

These plots clearly indicate that the 2 compartment model is a better fit compared to the one compartment model. We can look at selected sample of individual plots.

f = subject_fits(res_inspect_2cmp,
             separate = true, columns = 3, rows = 3)

We look at random set of 9 individuals here by indexing into generated plot

f[6]

There a lot of important plotting functions you can use for your standard model diagnostics. Please make sure to read the documentation for plotting and the tutorial associated with it. Below, we are checking the distribution of the empirical Bayes estimates.

f = empirical_bayes_dist(res_inspect_2cmp)[1]

f = empirical_bayes_vs_covariates(res_inspect_2cmp, categorical = [:DOSE])[1]
# increase the height of this plot with resolution scaling cc Michael

Clearly,our guess at tvka seems off-target. Let's try and estimate tvka instead of fixing it to 2

pkfit_2cmp_unfix_ka = fit(pk_2cmp,
                 pkpain_noplb,
                 init_params(pk_2cmp),
                 Pumas.FOCEI())

infer(pkfit_2cmp_unfix_ka)

Asymptotic inference results

Successful minimization:                      true

Likelihood approximation:               Pumas.FOCE
Log-likelihood value:                    1898.4797
Number of subjects:                            120
Number of parameters:         Fixed      Optimized
                                  0             11
Observation records:         Active        Missing
    CONC:                      1320              0
    Total:                     1320              0

----------------------------------------
        Estimate     SE      95.0% C.I.
----------------------------------------
tvcl     2.6191    NaN     [NaN; NaN]
tvv     11.378     NaN     [NaN; NaN]
tvvp     8.4531    NaN     [NaN; NaN]
tvq      1.3164    NaN     [NaN; NaN]
tvka     4.8926    NaN     [NaN; NaN]
Ω₁,₁     0.13243   NaN     [NaN; NaN]
Ω₂,₂     0.05967   NaN     [NaN; NaN]
Ω₃,₃     0.41581   NaN     [NaN; NaN]
Ω₄,₄     0.080682  NaN     [NaN; NaN]
Ω₅,₅     0.24996   NaN     [NaN; NaN]
σ_p      0.049098  NaN     [NaN; NaN]
----------------------------------------


Variance-covariance matrix could not be
evaluated. The random effects may be over-
parameterized. Check the coefficients for
variance estimates near zero.

compare_estimates(;pkfit_2cmp,pkfit_2cmp_unfix_ka)

11 rows × 3 columns

	parameter	pkfit_2cmp	pkfit_2cmp_unfix_ka
	String	Float64?	Float64?
1	tvcl	2.81378	2.61913
2	tvv	11.0046	11.3783
3	tvvp	5.53997	8.45305
4	tvq	1.51591	1.31635
5	tvka	2.0	4.89256
6	Ω₁,₁	0.102669	0.132432
7	Ω₂,₂	0.0607756	0.0596702
8	Ω₃,₃	1.20116	0.415811
9	Ω₄,₄	0.423493	0.0806819
10	Ω₅,₅	0.244732	0.24996
11	σ_p	0.0484049	0.0490976

Let's revaluate the goodness of fits and η distribution plots.

Not much change in the general gof plots

res_inspect_2cmp_unfix_ka = inspect(pkfit_2cmp_unfix_ka)
goodness_of_fit(res_inspect_2cmp_unfix_ka)[1]

But you can see a huge improvement in the ηka, (η₃) distribution which is now centered around zero

f = empirical_bayes_vs_covariates(res_inspect_2cmp_unfix_ka, categorical = [:DOSE])[1]
# increase the height of this plot with resolution scaling cc Michael

Finally looking at some individual plots for the same subjects as earlier

f = subject_fits(res_inspect_2cmp_unfix_ka,
             separate = true, columns = 3, rows = 3)

f[6]

The randomly sampled individual fits don't seem good in some individuals, but we can evaluate this via a vpc to see how to go about.

We can now perform a vpc to check. The default plots provide a 80% prediction interval and a 95% simulated CI (shaded area) around each of the quantiles

pk_vpc = vpc(pkfit_2cmp_unfix_ka, 200; observations = [:CONC],
             stratify_by = [:DOSE],
             ensemblealg=EnsembleThreads())

Data Quantiles

99×4 DataFrame
 Row │ DOSE    time     CONC       τ
     │ Int64?  Float64  Float64    Float64
─────┼─────────────────────────────────────
   1 │      5      0.5  0.247612       0.1
   2 │      5      1.0  0.223815       0.1
   3 │      5      1.5  0.200424       0.1
   4 │      5      2.0  0.17378        0.1
   5 │      5      2.5  0.153568       0.1
   6 │      5      3.0  0.132368       0.1
   7 │      5      4.0  0.101127       0.1
   8 │      5      5.0  0.0746744      0.1
  ⋮  │   ⋮        ⋮         ⋮         ⋮
  93 │     80      2.5  5.17378        0.9
  94 │     80      3.0  4.73682        0.9
  95 │     80      4.0  3.98179        0.9
  96 │     80      5.0  3.28932        0.9
  97 │     80      6.0  2.67354        0.9
  98 │     80      7.0  2.21093        0.9
  99 │     80      8.0  1.78044        0.9
                            84 rows omitted
Simulation Quantiles

99×6 DataFrame
 Row │ τ        time     DOSE    lower      middle     upper
     │ Float64  Float64  Int64?  Float64    Float64    Float64
─────┼───────────────────────────────────────────────────────────
   1 │     0.1      0.5       5  0.220358   0.249499   0.27681
   2 │     0.1      1.0       5  0.201199   0.226051   0.243749
   3 │     0.1      1.5       5  0.181384   0.202035   0.218417
   4 │     0.1      2.0       5  0.15927    0.178249   0.19316
   5 │     0.1      2.5       5  0.139358   0.157212   0.171493
   6 │     0.1      3.0       5  0.119327   0.137063   0.151284
   7 │     0.1      4.0       5  0.0853168  0.101561   0.117153
   8 │     0.1      5.0       5  0.0594752  0.0739719  0.0890552
  ⋮  │    ⋮        ⋮       ⋮         ⋮          ⋮          ⋮
  93 │     0.9      2.5      80  4.39338    4.75286    5.15241
  94 │     0.9      3.0      80  4.00087    4.35019    4.68268
  95 │     0.9      4.0      80  3.29853    3.61664    3.92773
  96 │     0.9      5.0      80  2.70039    2.97388    3.24849
  97 │     0.9      6.0      80  2.18019    2.4226     2.69547
  98 │     0.9      7.0      80  1.70069    1.97539    2.23599
  99 │     0.9      8.0      80  1.32364    1.62471    1.93912
                                                  84 rows omitted

f = vpc_plot(pk_2cmp,
        pk_vpc, rows=1, columns=1)

3-element Vector{AbstractPlotting.Figure}:
 Figure()
 Figure()
 Figure()

f[1]

f[2]

f[3]

The visual predictive check suggests that the model captures the data well across all dose levels.

This completes the introductory tutorial that covers a workflow for pharmacokinetic modeling. In part-2 of this tutorial we will use the results of the pharmacokinetic model and evaluate its impact on pain relief.

If you have questions regarding this tutorial, please post them on our discourse site.

---